Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by a Caputo fractional derivative, and the second order space derivative by a symmetric fractional derivative. First, a method of separating variables expresses the analytical solution of the TSS-FDE in terms of the Mittag--Leffler function. Second, we propose two numerical methods to approximate the Caputo time fractional derivative: the finite difference method; and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative and the second order space derivative by the symmetric fractional derivative. Firstly, a method of separating variables is used to express the analytical solution of the tss-fde in terms of the Mittag–Leffler function. Secondly, we propose two numerical methods to approximate the Caputo time fractional derivative, namely, the finite difference method and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results are presented to demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Response of tall buildings with symmetric setbacks under blast loading

This study explores three-dimensional nonlineardynamic responses of typical tall buildings with and without setbacks under blast loading. These 20 storey reinforced concrete buildings have been designed for normal (dead, live and wind)loads. The influence of the setbacks on the lateral load response due to blasts in terms of peak deflections, accelerations, inter-storey drift and bending moments at critical locations (including hinge formation) were investigated. Structural response predictions were performed with a commercially available three-dimensional finite element analysis programme using non-linear direct integration time history analyses. Results obtained for buildings with different setbacks were compared and conclusions made. The comparisons revealed that buildings have setbacks that protect the tower part above the setback level from blast loading show considerably better response in terms of peak displacement and interstorey drift, when compared to buildings without setbacks. Rotational accelerations were found to depend on the periods of the rotational modes. Abrupt changes in moments and shears are experienced near the levels of the setbacks. Typical twenty storey tall buildings with shear walls and frames that are designed for only normaln loads perform reasonably well, without catastrophic collapse, when subjected to a blast that is equivalent to 500 kg TNT at a standoff distance of 10 m.

Design of decoupling networks for circulant symmetric antenna arrays

Small element spacing in compact arrays results in strong mutual coupling between array elements. Performance degradation associated with the strong coupling can be avoided through the introduction of a decoupling network consisting of interconnected reactive elements. We present a systematic design procedure for decoupling networks of symmetrical arrays with more than three elements and characterized by circulant scattering parameter matrices. The elements of the decoupling network are obtained through repeated decoupling of the characteristic eigenmodes of the array, which allows the calculation of element values using closed-form expressions.

Single-layer decoupling networks for circulant symmetric arrays

Reduced element spacing in antenna arrays gives rise to strong mutual coupling between array elements and may cause significant performance degradation. These effects can be alleviated by introducing a decoupling network consisting of interconnected reactive elements. The existing design approach for the synthesis of a decoupling network for circulant symmetric arrays allows calculation of element values using closed-form expressions, but the resulting circuit configuration requires multilayer technology for implementation. In this paper, a new structure for the decoupling of circulant symmetric arrays of more than four elements is presented. Element values are no longer obtained in closed form, but the resulting circuit is much simpler and can be implemented on a single layer.

Plastic bending behaviour and section moment capacities of mono-symmetric LiteSteel Beams with web openings

Recently developed cold-formed LiteSteel beam (LSB) sections have found increasing popularity in residential, industrial and commercial buildings due to their light weight and cost-effectiveness. Currently, there is significant interest in the use of LSB sections as flexural members in floor joist systems, although they can be used as flexural and compression members in a range of building systems. The plastic bending behaviour and section moment capacity of LSB sections with web holes can be assumed to differ from those without, but have yet to be investigated. Hence, no appropriate design rules for determining the section moment capacity of LSB sections with web holes are yet available. This paper presents the results of an investigation of the plastic bending behaviour and section moment capacity of LSB sections with circular web holes. LSB sections with varying circular hole diameters and degrees of spacing were considered. The paper also describes the simplified finite element (FE) modelling technique employed in this study, which incorporates all of the significant behavioural effects that influence the plastic bending behaviour and section moment capacity of these sections. The numerical and experimental test results and associated findings are also presented.

Numerical method for predicting the elastic lateral distortional buckling moment of a mono-symmetric beam with web openings

When used as floor joists, the new mono-symmetric LiteSteel beam (LSB) sections require web openings to provide access for inspections and various services. The LSBs consist of two rectangular hollow flanges connected by a slender web, and are subjected to lateral distortional buckling effects in the intermediate span range. Their member capacity design formulae developed to date are based on their elastic lateral buckling moments, and only limited research has been undertaken to predict the elastic lateral buckling moments of LSBs with web openings. This paper addresses this research gap by reporting the development of web opening modelling techniques based on an equivalent reduced web thickness concept and a numerical method for predicting the elastic buckling moments of LSBs with circular web openings. The proposed numerical method was based on a formulation of the total potential energy of LSBs with circular web openings. The accuracy of the proposed method’s use with the aforementioned modelling techniques was verified through comparison of its results with those of finite strip and finite element analyses of various LSBs.

Member moment capacities of mono-symmetric LiteSteel beam floor joists with web openings

Recently developed cold-formed LiteSteel beam (LSB) sections have found increasing popularity in residential, industrial and commercial buildings due to their light weight and cost-effectiveness. Another beneficial characteristic is that they allow torsionally rigid rectangular flanges to be combined with economical fabrication processes. Currently, there is significant interest in the use of LSB sections as flexural members in floor joist systems. When used as floor joists, these sections require openings in the web to provide access for inspection and other services. At present, however, there is no design method available that provides accurate predictions of the moment capacities of LSBs with web openings. This paper presents the results of an investigation of the buckling and ultimate strength behaviour of LSB flexural members with web openings. A detailed fine element analysis (FEA)-based parametric study was conducted with the aim of developing appropriate design rules and making recommendations for the safe design of LSB floor joists. The results include the required moment capacity curves for LSB sections with a range of web opening combinations and spans and the development of appropriate design rules for the prediction of the ultimate moment capacities of LSBs with web openings.

The effect of surface tension and kinetic undercooling on a radially-symmetric melting problem

The addition of surface tension to the classical Stefan problem for melting a sphere causes the solution to blow up at a finite time before complete melting takes place. This singular behaviour is characterised by the speed of the solid-melt interface and the flux of heat at the interface both becoming unbounded in the blow-up limit. In this paper, we use numerical simulation for a particular energy-conserving one-phase version of the problem to show that kinetic undercooling regularises this blow-up, so that the model with both surface tension and kinetic undercooling has solutions that are regular right up to complete melting. By examining the regime in which the dimensionless kinetic undercooling parameter is small, our results demonstrate how physically realistic solutions to this Stefan problem are consistent with observations of abrupt melting of nanoscaled particles.

Principle components analysis with covariance matrix on regional differences for mortality and percentages of cancers in Shandong province [山东省恶性肿瘤死亡率和构成比地区差异协方差矩阵主成分分析]

Objective To explore the characteristics of regional distribution of cancer deaths in Shandong Province with the principle components analysis. Methods The principle components analysis with co-variance matrix for age-adjusted mortality rates and percentages of 20 types of cancer in 22 counties （cities） were carried out using SAS Software. Results Over 90% of the total information could be reflected by the top 3 principle components and the first principle component alone represented more than half of the overall regional variances. The first component mainly reflected the area differences of esophageal cancer. The second component mainly reflected the area differences of lung cancer， stomach cancer and liver cancer. The value of the first principal component scores showed a clear trend that the west areas possessed higher values and the east the lower values. Based on the top two components，the 22 counties （cities） could be divided into several geographical clusters. Conclusion The overall difference of regional distribution of cancers in Shandong is dominated by several major cancers including esophageal cancer， lung cancer， stomach cancer and liver cancer. Among them，esophageal cancer makes the largest contribution. If the range of counties （cities） analyzed could be further widened， the characteristics of regional distribution of cancer mortality would be better examined. Abstract in Chinese 目的 利用主成分分析探讨山东省恶性肿瘤死亡的地区分布特征. 方法 利用SAS软件对山东省22个县市区2004～2006午的20种恶性肿瘤标化死亡率和构成比分别进行协方差矩阵主成分分析. 结果 前3个主成分就反映了总体差异90%以上的信息,其中仅第1主成分就提供了总体差异一半以上的信息.第1主成分主要反映了食管癌的地区差异,第2主成分主要反映肺癌的地区差异,兼顾胃癌和肝癌.各地区第1主成分得分呈现西高东低的趋势,根据第1和第2主成分可以将调查地区分为若干类别,表现为明显的地理聚集性. 结论 山东省各地区恶性肿瘤死亡的总体差异主要取决于少数高发肿瘤,包括食管癌、肺癌、胃癌、肝癌等,其中以食管癌地位最为突出.如能进一步扩大分析范围,可更好地查明恶性肿瘤死亡的地区特征.